We noticed that the "processingOptions" has fields for "runBilling" and "collectPayment", but there is no field to select an existing "paymentMethodId" that could be used for this one purchase.

This was something that was available in the old SOAP API. It was dropped in the new REST API, although the legacy "actions" section of the REST API does still have some support for it.

Could the "paymentMethodId" be added to the new Orders feature?

This is an important feature for some of our customers who keep multiple credit cards on file. We are still using the older SOAP API but we want to adopt the new Orders functionaity because there are some big enhancements there, too.